Properties

An amnestic full and faithful functor is automatically an isocofibration, i.e. injective on objects: if UD′=UDU D' = U D, then there is some isomorphismf:D′→Df : D' \to D in 𝒟\mathcal{D} such that Uf=idUDU f = id_{U D}, but then we must have f=idUD′=idUDf = id_{U D'} = id_{U D}, so D′=DD' = D.

An amnestic isofibration has the following lifting property: for any object DD in 𝒟\mathcal{D} and any isomorphism f:C→UDf : C \to U D in 𝒞\mathcal{C}, there is a unique isomorphism f˜:C˜→D\tilde{f} : \tilde{C} \to D such that Uf˜=fU \tilde{f} = f. Indeed, if f˜′:C˜′→D\tilde{f}' : \tilde{C}' \to D were any other isomorphism such that Uf˜′=fU \tilde{f}' = f, then U(f˜−1∘f˜′)=idCU (\tilde{f}^{-1} \circ \tilde{f}') = id_C, so we must have f˜=f˜′\tilde{f} = \tilde{f}'.

If the compositeU∘KU \circ K is an amnestic functor, then KK is also amnestic.

Examples

Any strictly monadic functor is amnestic. Conversely, any monadic functor that is also an amnestic isofibration is necessarily strictly monadic.